The interaction between the spin angular momentum quantum numbers
of two electrons gives rise to states of so called different spin
multiplicity, such as singlet and doublet states.

The spin angular momentum of an electron may also
interact with the orbital angular momentum of the electron to split
the energies of the different levels, causing different transition
energies to be observed. This effect is known as spin-orbit
coupling.

The interaction of the spin angular momentum and the
orbital angular momentum gives rise to the total angular momentum:
when the spin and orbital angular momentum are parallel, the total
angular momentum is high.

For a given electron the total angular momentum, j,
is either l+s or l-s
(with s = 1/2 for the electron), and
the energy level corresponding l is
therefore split into two terms for the different values of j.

The energy of the state characterized by the quantum
numbers, l, s and j, is given by the expression:

Here, A is the spin-orbit
coupling constant, and the degree of coupling increases
as Z4 (where Z is the
atomic number).

The total angular momentum of a single electron
can also couple with the total angular momentum of other electrons,
such that the total angular momentum of an atom as a whole can
have a range of values, depending upon the exact occupation of
the orbitals. The spin, orbital, and total angular momentum for
an atom or ion in a given electronic configuration is expressed
as a term symbol.

In light atoms, the spin-orbit coupling in individual
electrons is weak, and instead the coupling of the spin angular
momenta of different electrons, or the coupling of the orbital
angular momenta of different atoms, is stronger. This type of
spin-orbit coupling is known as the Russell-Saunders
scheme, and gives rise to term symbols as shown, for a
species with two electrons in d orbitals, in the table below.

Russell
Saunders coupling: the d2 ion

When more than one valence electron
is present, interactions between the electrons result in
couplings between the quantum numbers for the individual
electrons. The quantum state of the overall ion depends
on the quantum states of the individual electrons.

The quantum state of the electron is determined by the
values of n (the principal quantum number), l (the orbital
angular momentum quantum number), ml (the magnetic
quantum number), and s (the spin quantum number).

There may be coupling between the spin angular momenta
of two electrons, spin-spin
coupling, the orbital angular momenta of two electrons,
orbit-orbit coupling, and
the spin and orbital angular momenta of the same electron,
spin-orbit coupling.

In the Russell-Saunders scheme, the case for the first
row transition elements, and in general for elements up
to atomic number 30, the magnitude of coupling is assumed
to be in the order:

spin-spin
coupling > orbit-orbit coupling > spin-orbit coupling

Spin-spin coupling:

The spin quantum number, S, for a system of electrons
is calculated from the spin quantum numbers, s1
and s2, for the separate electrons according
to

for the
d2 system,

Orbit-orbit coupling:

For two electrons with
orbital angular momentum quantum numbers l1
and l2, the total orbital angular momentum
quantum number, L, is

Spin-orbit coupling:

The total angular momentum quantum number, J, is obtained
by coupling the total spin and orbital angular momenta according
to:

Different values of S can have
different numbers of values of J, or different numbers of
levels. The number of levels
possible for a given S number is the multiplicity,
given by (2S + 1).

the d2 system
has multiplicity values

S = 1: (2S+1) = 3 (a triplet)

S = 0: (2S+1) = 1 (a singlet)

The information on the possible values of S,
L and J as summarized in the term symbol:

Not all terms are allowed, as some would require electrons
with the same spin to occupy the same orbital, in contravention
of the Pauli exclusion principle.

the d2
system has the possible terms:

3P, 3F, 1S,
1D, 1G

The relative order of the energies of these
terms is given by Hund's rules:

1) The most stable state is the one with the maximum
multiplicity

2) For a group of terms with the same multiplicity,
the one with the largest value of L lies lowest in energy.

the ground
state term for the d2 system is:

3F

An important consequence of term symbols is their
use to express the ranges of S, L, and J which may be involves
in allowed transitions between the levels the term symbols represent.
These allowed transitions may be summarized as a set of selection
rules:

ΔS = 0

ΔL = 0, +1,
-1 with Δl = +1, -1

ΔJ = 0, +1,
-1 but J = 0 to J = 0 is forbidden

These selection rules only apply in the Russell-Saunders
coupling scheme. In heavier atoms, the coupling between the spin
and orbital angular momentum of individual electrons is much stronger,
and only the total angular momentum, J,
is important. The selection rules based on the values of S and
L therefore do not hold. A better coupling scheme for the heavy
atoms is jj-coupling, where
the total angular momentum of each electron is calculated first,
and then these are coupled to give the overall total angular momentum
of the atom.